Robbie Willmott and Chris Zebroski both had chances to snatch the win for the Exiles but neither could beat Brian Jensen in the Bury goal.
Captain Craig Jones had two early chances for the Shakers but neither team could break the deadlock at Rodney Parade.
Danny Mayor burst down the left wing on two occasions in the opening minutes before feeding Welshman Jones but the Bury captain failed convert either of Mayor's driven crosses from six yards out.
Newport had their first chance of the game after 25 minutes through Zebroski, his low drive forced former Burnley keeper Brian Jensen into a reaction save down to his left. The Dane was tested again after Robbie Willmott curled a shot towards the top corner, but Jensen was equal to it.
Bury failed to test Ian McLoughlin in the County goal in a second half which the Welsh team dominated.